Being the user in question, let me give my point of view.
As others say, this is not a make or break issue for me. It is a preference. If someone exhibits a great deal of moderator related skill (mitigating conflict, ability to follow policies not to one's liking, participation in all sort of ways), I would mind less about who they are.
But I am using this site with my real name. And I am getting recognized for it in real life too. It is a good thing, in general. When I attend conferences, people tell me that their students told them about me, and my online exploits in answering their many questions; when visitors come to Jerusalem, I sometime hear from them that I help their students solve their homework assignments for good or bad.
This means that whatever image this website has in the eyes of people who do not participate in it, it will be associated with my image as well. Having a real life identity that I can associate with a person means not only that I can offer them a drink when I'm around, but also that I feel that I can trust them better not to screw up with my real life identity, simply because they would be screwing up their own as well.
In other SE cultures, programming, sci-fi, whatever, there are either long traditions of using a virtual persona, or not sufficient real life implications to benefit the user for using their real life persona. Naturally, on those sites moderators will be less obligated to use their real names, and since the users are anonymous it is also more likely that there might be actual spillover.
I'm not sure, and I can't quite know, how much real life spillage there has been to our current moderators, but I imagine it is less likely than other sites. (See below)
Finally, the issue of age, I am not saying that age is associated with maturity, but the younger the candidates, the higher probability that they are less-mature. I know because I felt quite mature at 17, and I can tell at 29 that I am still not as nearly as mature as I felt I was back then. And no, I didn't devolve my maturity, I just learned to be better aware of myself.
So I am less trusting of a 15 year old, especially since most teen candidates have little meta participation registered to them, making it virtually impossible for me to judge how mature they might be.
I just ran into Does a moderator candidate need to divulge the real name? which is a similar thread for the first election held on the site.
The arguments given by T.. (which, admittedly, at the time I supported) are similar to those that are given by HDE 226868 in their answer on this page.
But it has been four years now, and I don't recall hearing about a moderator being harassed online, and we had almost all the moderators use their real names (or easily trackable, anyway). Which means that while the concern was valid, it's not as bad as expected. (See below.)
So I can't stand by that argument anymore. I don't feel that using anonymity as a shield is beneficial.
It was pointed out in the comments here by one of the current moderators, that he was "email bombed" once, to which he responded by blocking the address and moving on.
That is not an actual consequence. This is a consequence of publishing your email on the web, including inside papers on arXiv. It means people can decide to harass you in email. To wit, I was acknowledged for some advice in a recent paper (by a user of this site, whose emails were certainly not harassment, and I am not sure if he would have found me without me using my real name), and several days after that paper was posted to arXiv, I began being harassed by a well-known crank.
It is true that I was related to that paper because I used my real name, but who's to say that if I weren't doing so, that user wouldn't have found me anyway through other means? It's hard to judge. But it does show that being mentioned in a paper can get you into the bulk spam-mailing list of arbitrary cranks.
Both myself and Andres Caicedo recount in comments on another answer that we were "harassed" by users either asking us about closure and edits, or asking questions. I don't find that to be distressing. Neither of us is a moderator, and while these might be arguments in favor of staying anonymous as much as you possibly can, and surely there are more arguments for that, I don't find them compelling as arguments for moderators being anonymous.
Having more power means that to establish trust you need more transparency. I am less likely to trust my personal information to someone who is not trusting me with their. And moderators do have access to that sort of information, and more. What can I do about that? Hide my identity, I don't want to do that. Just like anonymous users might not wish to reveal their secret identity to me, I don't want to hide mine. They are the Batmen of the site, and I'm Tony Stark, Iron Man.
What do I count as actual problem in this context? Repeated harassment which is not evadable by a few simple clicks (and I'm not talking about a killswitch for your internet service!). I prefer not to give any ideas here, though. For obvious reasons.
Finally, let me apologize for this mess of an edit. I feel it's far less coherent than the previous parts and for that I am truly sorry.